Polytope of Type {2,4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,84}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11397)
Rank : 4
Schlafli Type : {2,4,84}
Number of vertices, edges, etc : 2, 4, 168, 84
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,42}*672b
   4-fold quotients : {2,4,21}*336
   7-fold quotients : {2,4,12}*192c
   14-fold quotients : {2,4,6}*96c
   28-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3, 89)(  4, 90)(  5, 87)(  6, 88)(  7, 93)(  8, 94)(  9, 91)( 10, 92)
( 11, 97)( 12, 98)( 13, 95)( 14, 96)( 15,101)( 16,102)( 17, 99)( 18,100)
( 19,105)( 20,106)( 21,103)( 22,104)( 23,109)( 24,110)( 25,107)( 26,108)
( 27,113)( 28,114)( 29,111)( 30,112)( 31,117)( 32,118)( 33,115)( 34,116)
( 35,121)( 36,122)( 37,119)( 38,120)( 39,125)( 40,126)( 41,123)( 42,124)
( 43,129)( 44,130)( 45,127)( 46,128)( 47,133)( 48,134)( 49,131)( 50,132)
( 51,137)( 52,138)( 53,135)( 54,136)( 55,141)( 56,142)( 57,139)( 58,140)
( 59,145)( 60,146)( 61,143)( 62,144)( 63,149)( 64,150)( 65,147)( 66,148)
( 67,153)( 68,154)( 69,151)( 70,152)( 71,157)( 72,158)( 73,155)( 74,156)
( 75,161)( 76,162)( 77,159)( 78,160)( 79,165)( 80,166)( 81,163)( 82,164)
( 83,169)( 84,170)( 85,167)( 86,168)(171,257)(172,258)(173,255)(174,256)
(175,261)(176,262)(177,259)(178,260)(179,265)(180,266)(181,263)(182,264)
(183,269)(184,270)(185,267)(186,268)(187,273)(188,274)(189,271)(190,272)
(191,277)(192,278)(193,275)(194,276)(195,281)(196,282)(197,279)(198,280)
(199,285)(200,286)(201,283)(202,284)(203,289)(204,290)(205,287)(206,288)
(207,293)(208,294)(209,291)(210,292)(211,297)(212,298)(213,295)(214,296)
(215,301)(216,302)(217,299)(218,300)(219,305)(220,306)(221,303)(222,304)
(223,309)(224,310)(225,307)(226,308)(227,313)(228,314)(229,311)(230,312)
(231,317)(232,318)(233,315)(234,316)(235,321)(236,322)(237,319)(238,320)
(239,325)(240,326)(241,323)(242,324)(243,329)(244,330)(245,327)(246,328)
(247,333)(248,334)(249,331)(250,332)(251,337)(252,338)(253,335)(254,336);;
s2 := (  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)( 13, 24)
( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)( 33, 60)
( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)( 41, 80)
( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)
( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)( 96,109)
( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)(116,145)
(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)(124,165)
(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)(132,157)
(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)(140,149)
(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)(176,281)
(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)(184,273)
(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)(192,265)
(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)(200,313)
(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)(208,333)
(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)(216,325)
(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)(224,317)
(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)(232,309)
(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)(240,301)
(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)(248,293)
(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);;
s3 := (  3,231)(  4,234)(  5,233)(  6,232)(  7,227)(  8,230)(  9,229)( 10,228)
( 11,251)( 12,254)( 13,253)( 14,252)( 15,247)( 16,250)( 17,249)( 18,248)
( 19,243)( 20,246)( 21,245)( 22,244)( 23,239)( 24,242)( 25,241)( 26,240)
( 27,235)( 28,238)( 29,237)( 30,236)( 31,203)( 32,206)( 33,205)( 34,204)
( 35,199)( 36,202)( 37,201)( 38,200)( 39,223)( 40,226)( 41,225)( 42,224)
( 43,219)( 44,222)( 45,221)( 46,220)( 47,215)( 48,218)( 49,217)( 50,216)
( 51,211)( 52,214)( 53,213)( 54,212)( 55,207)( 56,210)( 57,209)( 58,208)
( 59,175)( 60,178)( 61,177)( 62,176)( 63,171)( 64,174)( 65,173)( 66,172)
( 67,195)( 68,198)( 69,197)( 70,196)( 71,191)( 72,194)( 73,193)( 74,192)
( 75,187)( 76,190)( 77,189)( 78,188)( 79,183)( 80,186)( 81,185)( 82,184)
( 83,179)( 84,182)( 85,181)( 86,180)( 87,315)( 88,318)( 89,317)( 90,316)
( 91,311)( 92,314)( 93,313)( 94,312)( 95,335)( 96,338)( 97,337)( 98,336)
( 99,331)(100,334)(101,333)(102,332)(103,327)(104,330)(105,329)(106,328)
(107,323)(108,326)(109,325)(110,324)(111,319)(112,322)(113,321)(114,320)
(115,287)(116,290)(117,289)(118,288)(119,283)(120,286)(121,285)(122,284)
(123,307)(124,310)(125,309)(126,308)(127,303)(128,306)(129,305)(130,304)
(131,299)(132,302)(133,301)(134,300)(135,295)(136,298)(137,297)(138,296)
(139,291)(140,294)(141,293)(142,292)(143,259)(144,262)(145,261)(146,260)
(147,255)(148,258)(149,257)(150,256)(151,279)(152,282)(153,281)(154,280)
(155,275)(156,278)(157,277)(158,276)(159,271)(160,274)(161,273)(162,272)
(163,267)(164,270)(165,269)(166,268)(167,263)(168,266)(169,265)(170,264);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(338)!(1,2);
s1 := Sym(338)!(  3, 89)(  4, 90)(  5, 87)(  6, 88)(  7, 93)(  8, 94)(  9, 91)
( 10, 92)( 11, 97)( 12, 98)( 13, 95)( 14, 96)( 15,101)( 16,102)( 17, 99)
( 18,100)( 19,105)( 20,106)( 21,103)( 22,104)( 23,109)( 24,110)( 25,107)
( 26,108)( 27,113)( 28,114)( 29,111)( 30,112)( 31,117)( 32,118)( 33,115)
( 34,116)( 35,121)( 36,122)( 37,119)( 38,120)( 39,125)( 40,126)( 41,123)
( 42,124)( 43,129)( 44,130)( 45,127)( 46,128)( 47,133)( 48,134)( 49,131)
( 50,132)( 51,137)( 52,138)( 53,135)( 54,136)( 55,141)( 56,142)( 57,139)
( 58,140)( 59,145)( 60,146)( 61,143)( 62,144)( 63,149)( 64,150)( 65,147)
( 66,148)( 67,153)( 68,154)( 69,151)( 70,152)( 71,157)( 72,158)( 73,155)
( 74,156)( 75,161)( 76,162)( 77,159)( 78,160)( 79,165)( 80,166)( 81,163)
( 82,164)( 83,169)( 84,170)( 85,167)( 86,168)(171,257)(172,258)(173,255)
(174,256)(175,261)(176,262)(177,259)(178,260)(179,265)(180,266)(181,263)
(182,264)(183,269)(184,270)(185,267)(186,268)(187,273)(188,274)(189,271)
(190,272)(191,277)(192,278)(193,275)(194,276)(195,281)(196,282)(197,279)
(198,280)(199,285)(200,286)(201,283)(202,284)(203,289)(204,290)(205,287)
(206,288)(207,293)(208,294)(209,291)(210,292)(211,297)(212,298)(213,295)
(214,296)(215,301)(216,302)(217,299)(218,300)(219,305)(220,306)(221,303)
(222,304)(223,309)(224,310)(225,307)(226,308)(227,313)(228,314)(229,311)
(230,312)(231,317)(232,318)(233,315)(234,316)(235,321)(236,322)(237,319)
(238,320)(239,325)(240,326)(241,323)(242,324)(243,329)(244,330)(245,327)
(246,328)(247,333)(248,334)(249,331)(250,332)(251,337)(252,338)(253,335)
(254,336);
s2 := Sym(338)!(  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)
( 13, 24)( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)
( 33, 60)( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)
( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)
( 57, 64)( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)
( 96,109)( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)
(116,145)(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)
(124,165)(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)
(132,157)(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)
(140,149)(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)
(176,281)(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)
(184,273)(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)
(192,265)(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)
(200,313)(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)
(208,333)(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)
(216,325)(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)
(224,317)(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)
(232,309)(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)
(240,301)(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)
(248,293)(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);
s3 := Sym(338)!(  3,231)(  4,234)(  5,233)(  6,232)(  7,227)(  8,230)(  9,229)
( 10,228)( 11,251)( 12,254)( 13,253)( 14,252)( 15,247)( 16,250)( 17,249)
( 18,248)( 19,243)( 20,246)( 21,245)( 22,244)( 23,239)( 24,242)( 25,241)
( 26,240)( 27,235)( 28,238)( 29,237)( 30,236)( 31,203)( 32,206)( 33,205)
( 34,204)( 35,199)( 36,202)( 37,201)( 38,200)( 39,223)( 40,226)( 41,225)
( 42,224)( 43,219)( 44,222)( 45,221)( 46,220)( 47,215)( 48,218)( 49,217)
( 50,216)( 51,211)( 52,214)( 53,213)( 54,212)( 55,207)( 56,210)( 57,209)
( 58,208)( 59,175)( 60,178)( 61,177)( 62,176)( 63,171)( 64,174)( 65,173)
( 66,172)( 67,195)( 68,198)( 69,197)( 70,196)( 71,191)( 72,194)( 73,193)
( 74,192)( 75,187)( 76,190)( 77,189)( 78,188)( 79,183)( 80,186)( 81,185)
( 82,184)( 83,179)( 84,182)( 85,181)( 86,180)( 87,315)( 88,318)( 89,317)
( 90,316)( 91,311)( 92,314)( 93,313)( 94,312)( 95,335)( 96,338)( 97,337)
( 98,336)( 99,331)(100,334)(101,333)(102,332)(103,327)(104,330)(105,329)
(106,328)(107,323)(108,326)(109,325)(110,324)(111,319)(112,322)(113,321)
(114,320)(115,287)(116,290)(117,289)(118,288)(119,283)(120,286)(121,285)
(122,284)(123,307)(124,310)(125,309)(126,308)(127,303)(128,306)(129,305)
(130,304)(131,299)(132,302)(133,301)(134,300)(135,295)(136,298)(137,297)
(138,296)(139,291)(140,294)(141,293)(142,292)(143,259)(144,262)(145,261)
(146,260)(147,255)(148,258)(149,257)(150,256)(151,279)(152,282)(153,281)
(154,280)(155,275)(156,278)(157,277)(158,276)(159,271)(160,274)(161,273)
(162,272)(163,267)(164,270)(165,269)(166,268)(167,263)(168,266)(169,265)
(170,264);
poly := sub<Sym(338)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope